Platonic polyhedra models have been readily available for a long time, but the Archimedean solids and stellations are seldom seen outside of museums. Their major attraction is their complexity, but this has meant great difficulty in their production. The table emphasizes the relative simplicity of the Platonic solids and the fascinating complexity of the Archimedean and the stellations.
TABLE 1 __________________________________________________________________________ POLYHEDRON FACES Triangles Squares Pentagons Hexagons __________________________________________________________________________ PLATONIC -- (Regular - Equal, regular faces) 1 TET Tetrahedron 4 2 CUBE Cube 6 3 OCT Octahedron 8 4 DOD Dodecahedron 12 5 ICOS Icosahedron 20 ARCHIMEDEAN -- (Uniform - Regular faces, vertices alike) 6 TT Truncated TET 4 4 7 TO Trunc. OCT 6 8 8 CO Cuboctahedron 8 6 Octagons 9 TC Trunc. Cube 8 6 10 TCO Trunc. Cuboct. 12 8 6 11 RCO Rhombicuboct. 8 18 12 SNC Snub Cube 32 6 13 TI Trunc. Icosahed. 12 20 14 ID Icosidodecahed. 20 12 Decagons 15 TD Trunc. Dodecahed. 20 12 16 TID Trunc. Icosidod. 30 20 12 17 RID Rhombicosidod. 20 30 12 18 SND Snub Dodecahed. 80 12 STELLATIONS -- (Uniform, non-convex) 19 GSD Great Stell. DOD 60 20 GD Great DOD 60 21 SSD Small Stell. DOD 60 22 GI Great ICOS 180 __________________________________________________________________________
On page 381 of the Mathematische Zeitschrift, vol.46 (1940) H. S. M. Coxeter states "that a polyhedron is edge-reflexible if all its edges are perpendicularly bisected by planes of symmetry", that "the vertices of any edge-reflexible polyhedron can be constructed . . . by reflections of a single point . . . by means of the polyhedral kaleidoscope, . . . a set of three plane mirrors, suitably inclined to one another", and that "the only uniform polyhedra which are not edge-reflexible are the snub cube, the snub dodecahedron and the antiprisms".
In his Chapter 3.2, "The Icosahedral Kaleidoscope" of Regular Complex Polytopes, MacMillan, New York, 1976, Coxeter describes a three-mirror device for recreating visually the ICOS, DOD, and some stellations. Its sharp vertex angles (about 21, 32, and 37 degrees) produce an undesirably small asymmetric field of view and only an, unconvincing linear outline of the solid, created by a line segment which is difficult to change.
In Solids,Geometric on p.861 of Vol.20 of the Encyclopedia Britannica, 1969, he states that as the TC can be formed by truncation of the CUBE, so the CO, TO, and OCT (the rest of the CUBE-OCT or C-O family) can be formed similarly by successive truncation, that this process can be reversed, OCT to CUBE (O-C family), and that this system applies similarly to the ICOS, TI, ID, TD, DOD (I-D family) and its reverse (D-I family).